STA 6467: Probability Theory II
Spring 2003
Problems for Assignment 3
Turn in the first three problems. The remaining problems are optional problems that you may wish to
try, but they will not be collected. Other optional problems from Section 22 of Billingsley are listed on the
course web page.
1. Let {Xn}be independent and identically distributed with mean 0and variance 1. Then we have for
every x:
Pmax
1≤k≤nSk≥x≤2PSn≥x−√2n
Hint: Let τbe the first time kat which Sk≥xand show that
PSn≥x−√2n≥
n
X
k=1
Pτ=k∩hSn−Sk≥ −√2ni.
2. For any sequence of random variables {Xn}and Sn=Pn
k=1 Xk, it is easy to show that
Sn
n
a.s.
−→ 0implies max1≤k≤n|Sk|
n
a.s.
−→ 0
Suppose that X1, X2, . . . are independent. Show in this case that
Sn
n
P
→0implies max1≤k≤n|Sk|
n
P
→0.
Hint: Use Etemadi’s inequality.
3. Let Xn≥0be independent for n≥1. The following are equivalent:
(i) P∞
n=1 Xn<∞a.s..
(ii) P∞
n=1P(Xn>1) + E[XnI[Xn≤1]]<∞.
(iii) P∞
n=1 EXn/(1 + Xn)<∞.
4. Let {Xn, n ≥1}be independent. Then for any α,
Pmax
1≤k≤nSk≥2α×min
1≤j≤nP
Sn−Sj
≤α≤P|Sn| ≥ α.
Hint: Consider sets of the form Sj<2α, 1≤j≤k−1, Sk≥2α, |Sn−Sk|< α.
5. For arbitrary {Xn}, if
X
n
E(|Xn|)<∞,
then PnXnconverges absolutely a.s.
